arXiv:0710.2239v1 [quant-ph] 11 Oct 2007
LYCEN 2007-19
Magnetic fields in
noncommutative quantum mechanics
Dedicated to the memory of Julius Wess1
François Delduc∗ , Quentin Duret† ,
François Gieres 2, ‡, Matthieu Lefrançois‡
∗
Université de Lyon, Laboratoire de Physique, UMR 5672
École Normale Supérieure de Lyon, 46 allée d’Italie,
F-69364 Lyon cedex 07 (France)
†
Laboratoire de Physique Théorique et Hautes Énergies, UMR 7589
(CNRS / Université P. et M. Curie Paris 6 / UniversitéD. Diderot Paris 7),
Tour 24-25, 5eme étage, BP 126, 4 place Jussieu,
F-75252 Paris Cedex 05 (France)
‡
Université de Lyon, Institut de Physique Nucléaire de Lyon,
Université Lyon 1 and CNRS/IN2P3, Bat. P. Dirac,
4 rue Enrico Fermi, F-69622 Villeurbanne Cedex (France)
Abstract
We discuss various descriptions of a quantum particle on noncommutative space
in a (possibly non-constant) magnetic field. We have tried to present the basic facts
in a unified and synthetic manner, and to clarify the relationship between various
approaches and results that are scattered in the literature.
1
We dedicate these notes to the memory of Julius Wess whose scientific work was largely devoted to
the study of gauge fields and whose latest interests concerned physical theories on noncommutative space.
The fundamental and inspiring contributions of Julius to Theoretical Physics will always bear with us, but
his great kindness, his clear and enthusiastic presentations, and his precious advice will be missed by all
those who had the chance to meet him.
2
Corresponding author: gieres@ipnl.in2p3.fr
Contents
1 Introduction and overview
1
2 Reminder on a particle in a magnetic field in QM
2.1 General magnetic fields . . . . . . . . . . . . . . . . .
2.1.1 Hamiltonian involving a vector potential . . .
2.1.2 Commutation relations involving the magnetic
2.2 Constant magnetic field (Landau system) . . . . . . .
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3 Noncommutative classical mechanics
3.1 Poisson algebra and its representations . . . . . . .
3.1.1 Poisson brackets . . . . . . . . . . . . . . .
3.1.2 Transformation to canonical coordinates . .
3.1.3 Case of non-constant magnetic field . . . . .
3.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Poisson brackets involving the magnetic field
3.2.2 Hamiltonian involving a vector potential . .
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4 NCQM: representations of the algebra
4.1 Representations of the algebra . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 About the relation between the parameters θ and B . . . . . . . . . . . . .
9
9
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5 NCQM systems: generalities
5.1 About Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Remarks on the ordering problem . . . . . . . . . . . . . . . . . . . . . . .
5.3 Properties of energy spectra . . . . . . . . . . . . . . . . . . . . . . . . . .
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6 NCQM systems: operatorial approach
6.1 Commutation relations involving a constant magnetic field . . . . . . . . .
6.2 Minimal coupling to a (possibly non-constant) magnetic field . . . . . . . .
6.3 “Exotic” approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
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15
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7 NCQM systems: star product approaches
7.1 Scalar potentials . . . . . . . . . . . . . . . . . .
7.2 Magnetic fields . . . . . . . . . . . . . . . . . . .
7.2.1 Gauge potentials . . . . . . . . . . . . . .
7.2.2 Gauge potentials and Seiberg-Witten map
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8 Summary and concluding remarks
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25
A Noncommuting coordinates in the truncated Landau problem (Peierls’
substitution)
27
1
Introduction and overview
What noncommutativity is about: Let us assume that the components X̂1 , . . . , X̂d of
the quantum mechanical position operator in d-dimensional space do not commute with
each other, but rather satisfy the commutation relations [X̂i , X̂j ] = iθij 1l where θij = −θji
are real numbers (at least one of which is non-zero). The Cauchy-Schwarz inequality then
implies the uncertainty relations (∆ψ X̂i )(∆ψ X̂j ) ≥ 21 |θij |, i.e. the particle described by the
wave function ψ cannot be localized in a precise way. This uncertainty relation for the
position implies a certain fuzziness of points in space: one says that the space is fuzzy,
pointless or that it has a lattice, quantum or noncommutative structure [1]. Obviously, the
noncommutativity can only manifest itself if the configuration space is at least of dimension
two. Thus, it represents a deformation of the classical theory which is quite different from
the deformation by quantization which concerns all dimensions and which amounts to
introducing a cellular structure (parametrized by ~) in phase space.
History of the subject: The idea of a fuzzy configuration space looks quite interesting
for quantum field theories since it may help to avoid, or at least ameliorate, the problem
of short-distance singularities, i.e. ultraviolet divergences. Indeed3 , this argument has
already been put forward in 1930 by Heisenberg [4] and the message was successively
passed on to Peierls, Pauli [5], Oppenheimer and Snyder (who was a student of the latter).
In 1933, Peierls worked out a quantum mechanical application concerning a particle in a
constant magnetic field (see appendix) [6] and, in 1947, Snyder considered noncommuting
coordinates on space-time in order to discard the ultraviolet divergences in quantum field
theory without destroying the Lorentz covariance [7]. In the sequel, the latter work was not
further exploited due to the remarkable success of the renormalization scheme in quantum
electrodynamics. Recently, noncommuting coordinates have regained interest, in particular
in the framework of superstring theories and of quantum gravity. Roughly speaking, the
intrinsic length scale of the strings induces a noncommutative structure on space-time at
low scales. (For a review of these topics, and in particular of quantum field theory on
noncommutative space-time and its renormalization, see for instance references [8]-[11].)
Since quantum mechanics may be regarded as a field theory in zero spatial dimensions or
as a non-relativistic description of the one-particle sector of field theory, it is well suited
as a toy model for the introduction of noncommuting coordinates.
Definition of NCQM: Quantum mechanics on noncommutative configuration space is
generally referred to as noncommutative quantum mechanics (NCQM). Somewhat more
generally, one can consider the situation where the commutators for coordinates and momenta are non-canonical. After some precursory work dealing with noncommutative phase
space variables in quantum mechanics [12]-[15], NCQM has been defined in a simple and
direct manner using different approaches [16]-[20], and some basic physical models have
been studied [16]-[25]. In the sequel, various aspects of the subject have been further elaborated upon and the literature has grown to a considerable size over the last years4 . The
3
4
The history of the subject has been traced back by J. Wess and is reported upon in references [2, 3].
We apologize to the authors whose work is not explicitly cited here and we refer to [26] for a more
1
study of exactly solvable models of NCQM should hopefully lead to a better understanding
of some issues in noncommutative field theory. There are some potential physical applications which include condensed matter physics (quantum Hall effect, superfluidity,...) and
spinning particles. By considering classical Poisson brackets (instead of commutators) and
Hamilton’s equations of motion, one can also discuss noncommutative classical mechanics,
as well as applications thereof like magneto-hydrodynamics [27, 2].
About the physical interpretation and applications: Since a cellular structure in
configuration space is not observed at macroscopic scales, the noncommutativity parameters θij should only manifest themselves at a length
p scale which is quite small compared to
some basic length scale like the Planck length ~G/c3 [1]. (In the context of field theory,
one usually writes θij = Λ21 θ̃ij , where the θ̃ij are dimensionless and of order 1, so that
NC
ΛN C represents a characteristic energy scale for the noncommutative theory which is necessarily quite large.) Thus, noncommutativity of space may be related to gravity at very
short distances and NCQM may be regarded as a deformation of classical mechanics that
is independent of the deformation by quantization. From this point of view, a fully fledged
theory of quantum gravity should provide a fuller understanding of the noncommutativity
of space.
In the framework of noncommutative classical or quantum mechanics, the parameters
θij admit many close analogies with a constant magnetic field both from the algebraic and
dynamical points of view [24]. Such an effective magnetic field, which is necessarily quite
small, may play a role as primordial magnetic field in cosmological dynamics [28].
Though various physical applications of NCQM have been advocated (in particular for
quantum mechanical systems coupled to a constant magnetic field), the only experimental
signature of noncommuting spatial coordinates which is currently available appears to be
the approximate noncommutativity appearing in the Landau problem for the limiting case
of a very strong magnetic field – see appendix. (This is a realization of noncommutativity
which is analogous to the one in string theory).
Scope of the present paper: The essential features of NCQM can be exhibited by
considering the following three instances to which we limit ourselves in the present paper.
1. We focus on flat 2-dimensional space so that the antisymmetric matrix (θij ) is
given by θij = θεij where θ is a real number and εij are the components of the
antisymmetric tensor normalized by ε12 = 1.
2. We assume that the fundamental commutator algebra has the following form
(which is not the most general form that one can consider for NCQM [21]):
[X̂1 , X̂2 ] = iθ1l ,
[P̂1 , P̂2 ] = iB1l ,
[X̂i , P̂j ] = iδij 1l .
(1)
Here, the real constant B (which measures the noncommutativity of momenta) describes a constant magnetic field that is perpendicular to the X̂1 X̂2 -plane. In equations (1) and in the following, the variables X̂i , P̂j (which are to be viewed as selfadjoint Hilbert space operators) are denoted by a hat so as to distinguish them from
complete list of references.
2
the basic operators Xi , Pj of standard quantum mechanics which satisfy the Heisenberg algebra, i.e. the canonical commutation relations (CCR’s)
[X1 , X2 ] = 0 = [P1 , P2 ] ,
[Xi , Pj ] = i~δij 1l .
(2)
As we will see, one can express the operators X̂1 , X̂2 , P̂1 , P̂2 as linear combinations
of X1 , X2 , P1 , P2 with coefficients which depend on the noncommutativity parameters
θ and B (that describe the deviation from the canonical commutators). We note
that the parameters θ and B in equations (1) are not necessarily independent of
each other and we will see that their interplay yields some particularly interesting
results [20, 22, 23, 24].
3. We are interested in the case of a possibly non-constant magnetic field, i.e. B
possibly depending on the spatial coordinates.
Short preview of results: The main results concerning the representations of the algebra
~ˆ P~ˆ ) may be summarized as follows.
(1) and the spectra of Hamiltonian operators H(X,
• For B 6= 1/θ (e.g. for B = 0), there exists a linear invertible transformation relating
the operators X̂i , P̂j to operators Xi , Pj satisfying canonical commutation relations.
By contrast, if B and θ are related by B = 1/θ, the transformation from X̂i , P̂j to
Xi , Pj is no longer invertible. The singular behavior is also reflected by the fact that
the so-called canonical limit (θ, B) → (0, 0) does not exist if B = 1/θ. Moreover,
in this case the four-dimensional phase space degenerates to a two-dimensional one
in the sense that for any irreducible representation of the commutator algebra, the
representation of the X̂i alone becomes irreducible.
• While the Jacobi identities for the algebra (1) involving the constant magnetic field
B are trivially satisfied, they are violated for a non-constant field B if θ 6= 0. Thus,
contrary to standard quantum mechanics, such a magnetic field forces us to introduce
the interaction into the Hamiltonian by means of a vector potential and a minimal
coupling. The noncommutativity of configuration space then leads to a non-Abelian
gauge structure and thus to self-interactions of the gauge potential which do not exist
in standard quantum mechanics.
~ˆ P~ˆ ) = 1 P~ˆ 2 + V (X̂1 , X̂2 )
• The spectrum of a typical Hamiltonian operator like H(X,
2m
depends on the ordering of the noncommuting operators X̂1 , X̂2 . Apart from some
special cases, the energy spectrum for θ 6= 0 differs notably from the one for θ = 0.
Non-polynomial potentials V (X̂1 , X̂2 ) become non-local when expressed in terms of
canonical coordinates X1 , X2 . Depending on the chosen parametrization of the phase
space variables X̂i , P̂j in terms of canonical variables Xi , Pj , the Hamiltonian may
take different disguises. However, in many instances, the parameter θ resembles a
constant magnetic field.
3
Outline of the presentation: To start with, we recall the description of a particle in a
magnetic field within standard quantum mechanics since it represents a useful guideline for
noncommutativity in quantum mechanics and a starting point for generalizations. Section 3
deals with noncommutative classical mechanics. In section 4, we discuss the representations
of the commutator algebra of NCQM. The remainder of the text is devoted to the study
of physical models of NCQM: we discuss some general properties of Hamiltonian operators
and of their spectra, and then treat in detail the problem of a (possibly non-constant)
magnetic field. The text concludes with a summary and some remarks.
2
Reminder on a particle in a magnetic field in QM
In non-relativistic quantum mechanics in Rd , the coordinates Xi and momenta Pj satisfy
the CCR’s. It is worthwhile recalling that the Schrödinger representation on L2 (Rd , dd x)
(i.e. Pj = ~i ∂x∂ j and Xi = operator of multiplication by the real variable xi ) is the only
possible irreducible realization of the CCR’s up to unitary equivalence5 [29, 30].
2.1
General magnetic fields
Concerning the algebra of NCQM and its representations, it is useful to have in mind
the standard quantum mechanical treatment [31] of a charged particle in three dimensions
~ x ) deriving from a vector potential A(~
~ x ). In the
which is coupled to a magnetic field B(~
next two paragraphs, we recall the two different, though equivalent descriptions of the
problem under consideration.
2.1.1
Hamiltonian involving a vector potential
We start from the coordinates Xi and the components Pj of the canonical momentum P~
satisfying the CCR’s (2). If the spin of the particle is not taken into account, the interaction
~ = rot
~ is simply described by means of the so-called minimal
~ A
with the magnetic field B
coupling, i.e. by the Hamiltonian
H=
2.1.2
1 ~ e~ 2
(P − A ) .
2m
c
(3)
Commutation relations involving the magnetic field
The Hamiltonian (3) is quadratic in the components of the kinematical (gauge invariant)
~ ≡ P~ − e A:
~
[32] momentum Π
c
1 ~2
H=
Π .
(4)
2m
5
In the relation [Xk , Pk ] = i~1l at least one of the self-adjoint operators Xk , Pk has to be unbounded,
henceforth it is not defined on all of Hilbert space, but only on some subspace. For this reason, one usually
exponentiates all of these operators so as to obtain unitary (i.e. bounded) operators satisfying Weyl’s form
of the CCR’s: the cited classification theorem only makes sense for this form of the CCR’s.
4
1 ~ 2
P with P~ replaced
Thus, it has the same form as the free particle Hamiltonian H0 ≡ 2m
~ The momentum variables Πi do not commute with each other, rather they satisfy
by Π.
the non-canonical commutation relations
e
where Bij ≡ ∂i Aj − ∂j Ai = εijk Bk ,
(5)
[Πi , Πj ] = i~ Bij ,
c
[Xi , Xj ] = 0 ,
[Xi , Πj ] = i~ δij 1
for i, j ∈ {1, 2, 3} .
In other words, when starting from the description of a free particle, the coupling of the
particle to a magnetic field can simply be described by replacing the canonical momentum
~ whose components have a
P~ in the free Hamiltonian by the kinematical momentum Π
non-vanishing commutator: it is the magnetic field which measures this noncommutativity.
We note that the Jacobi identities for the quantum algebra (5) are satisfied for any
~ x ) and, in particular, one has [Xi , [Πj , Πk ]] + circular permutations = 0.
field strength B(~
~ 6= ~0, the transformation (X,
~ P~ ) 7→ (X,
~ Π
~ ) is not unitary since it modifies the
For B
commutators (2).
2.2
Constant magnetic field (Landau system)
In the particular case where the magnetic field is constant, the coordinate axes can be
~ = B~e3 . The three-dimensional problem then reduces to a twooriented such that B
dimensional one, namely the problem of a particle in the x1 x2 -plane subject to a vector
~ = (A1 , A2 ) such that B = ∂1 A2 − ∂2 A1 : the Hamiltonian then reads as
potential A
1
(Π21 + Π22 ) ,
(6)
2m
1
if one discards the term 2m
P32 describing a free particle motion along the x3 -direction. The
present physical system (i.e. a charged particle in a plane subject to a constant magnetic
field that is perpendicular to this plane) is often referred to as the Landau system. For
this problem, convenient choices of the vector potential are the symmetric gauge (A1 , A2 ) =
(− B2 x2 , B2 x1 ) and the Landau gauge, i.e. (A1 , A2 ) = (0, Bx1 ) or (A1 , A2 ) = (−Bx2 , 0).
The energy spectrum of the Landau system can be obtained by a simple argument [12].
~ = (Π1 , Π2 ) and the latter variables satisfy the
In fact, the Hamiltonian only depends on Π
e
commutation relation [Π1 , Π2 ] = i~ c B1l, or equivalently
c
[Q, Π2 ] = i~ 1l
with Q ≡
Π1 .
(7)
eB
Thus, the operators Q and Π2 can be viewed as canonically conjugate variables in terms
of which the Hamiltonian reads as
|eB|
1 2 1
Π2 + mωB2 Q2
with ωB ≡
.
(8)
H=
2m
2
mc
Accordingly, the Landau system is equivalent to a linear harmonic oscillator and its eigenvalues (the so-called Landau levels) are given by
H=
En = ~
|eB|
1
(n + )
mc
2
with n = 0, 1, 2, . . .
5
(9)
Each of these energy levels is infinitely degenerate (with respect to the momentum eigenvalues p1 ∈ R) which reflects the fact that the Landau system is actually two-dimensional.
3
Noncommutative classical mechanics
Instead of an algebra of commutators, one can consider its classical analogon [33, 15]
involving Poisson brackets {·, ·}PB of functions depending on the real variables x1 , x2 , p1 , p2 .
3.1
3.1.1
Poisson algebra and its representations
Poisson brackets
The classical algebra associated to (1) reads as [33, 28]
{x1 , x2 }PB = θ ,
{p1 , p2 }PB = B ,
{xi , pj }PB = δij .
(10)
The so-called “exotic” algebra [15, 23, 25] amounts to modifying these brackets by a factor
1/κ where
κ ≡ 1 − Bθ ,
(11)
i.e.
B
δij
θ
,
{p1 , p2 }PB = ,
{xi , pj }PB =
.
(12)
κ
κ
κ
The expression κ is a characteristic parameter for the algebraic system under study and
will frequently reappear in the sequel.
If we denote the phase space coordinates collectively by (ξ I )I=1,...,4 ≡ (x1 , x2 , p1 , p2 ),
the
bracket of two functions F and G on phase space reads as {F, G}PB ≡
P4 Poisson
IJ ∂F ∂G
Ω
, so that {ξ I , ξ J }PB = ΩIJ , and the symplectic 2-form is given by
I,J=1
∂ξ I ∂ξ J
{x1 , x2 }PB =
ω=
4
1 X
ωIJ dξ I ∧ dξ J ,
2 I,J=1
(13)
with ωIJ ≡ (Ω−1 )IJ . The symplectic matrix Ω ≡ (ΩIJ ) for the standard and exotic algebras
(10) and (12) is respectively given by


0
θ
1
0
 −θ 0
1
0
1 
,
(14)
Ωexotic = Ωstandard .
Ωstandard = 
 −1 0
0 B 
κ
0 −1 −B 0
Note that det Ωstandard = κ2 , hence det Ωexotic = κ−2 . For θ = 0 or B = 0, the matrices
Ωstandard and Ωexotic coincide with each other. The brackets (12) defining the exotic algebra
diverge for κ → 0, but not so the associated symplectic form. For the “standard” algebra
(10), one has the reversed situation, i.e. one does not have a well defined symplectic form.
6
3.1.2
Transformation to canonical coordinates
By definition, the phase space R4 equipped with a non-degenerate, antisymmetric bilinear
form (13) is a symplectic vector space [34, 35]. (The condition of non-degeneracy is equivalent to det (ωIJ ) 6= 0.) For such a space, one can find a change of basis (i.e. an invertible
linear transformation of coordinates, (ξ I) 7→ (ξ ′I )) such that the matrix (ωIJ ) goes over
0 1l2
. This result [34, 35], which is also
into the canonical (normal) form J =
−1l2 0
referred to as the linear Darboux theorem, represents a special case of the Darboux
theorem which applies to general phase space manifolds. The explicit form of the linear
transformation (ξ I ) 7→ (ξ ′I ) for the algebra (1) will be discussed in section 4 in the context
of quantum mechanics. Here, we only emphasize that this transformation is not canonical
(in the sense of Hamiltonian mechanics) since it modifies the Poisson brackets.
3.1.3
Case of non-constant magnetic field
Let us consider the Poisson algebra (10) with constant parameter θ 6= 0 and with a parameter B which depends on ~x (but not on p~ ). Then,
{xi , {p1 , p2 }}PB + circular permutations = θεij
∂B
∂xj
for i = 1, 2 ,
i.e. the Jacobi identities are only satisfied if B is a constant magnetic field. By contrast,
the exotic algebra (12) with constant parameter θ 6= 0 allows for a ~x-dependent B-field
since such a field is compatible with the corresponding Jacobi identities [25]. Thus, in
the noncommutative plane, the coupling of a particle to a non-constant magnetic field
can be achieved by the exotic approach, but not by the standard one. For θ = 0, the two
approaches coincide with each other. We will come back to the exotic algebra in section 6.3.
3.2
Dynamics
The time evolution of a function F on phase space is given by Hamilton’s equation of
motion Ḟ = {F, H}PB where H is the Hamiltonian function. We will only consider the
case of a constant magnetic field.
3.2.1
Poisson brackets involving the magnetic field
The dynamics based on the Poisson algebra (10) has been studied in references [33, 28, 36].
1
For the Hamiltonian H = 2m
~p 2 + V (x1 , x2 ), one finds the equation of motion
∂V
d ∂V
mẍi = −κ
for i ∈ {1, 2} ,
(15)
+ Bεij ẋj + mθεij
∂xi
dt ∂xj
where κ ≡ 1 − Bθ. If the singular limit κ → 0 is taken, the first (Newton-like) force on
the right-hand-side vanishes and the number of dynamical degrees of freedom is reduced
by half. (This result is related to the Peierls substitution discussed in the appendix.)
7
Some specific potentials, e.g. the one of the anisotropic harmonic oscillator, are elaborated upon in reference [28]. Here, we will only discuss the case V = 0, i.e. the dynamics
of a charged particle (of unit charge) in the noncommutative plane subject to a constant
magnetic field which is perpendicular to this plane. The equation of motion (15) then
reduces to
mẍi = Bεij ẋj
for i ∈ {1, 2} .
(16)
Thus, we have exactly the same equation of motion as for θ = 0. Indeed, the fact that
H does not depend on the coordinates xi implies that the noncommutativity of these
coordinates does not manifest itself in the classical trajectories, though it affects some
physical quantities (like the density of states per unit area – see section 4 below for the
quantum theory).
3.2.2
Hamiltonian involving a vector potential
Let us now consider the Poisson algebra (10) with B = 0 and introduce a constant magnetic
~ = (A1 , A2 ), i.e. ∂1 A2 −∂2 A1 = B. The equations
field B deriving from a vector potential A
1
~ )2 then read
of motion following from the minimally coupled Hamiltonian H = 2m
(~p − A
as [33]
mẋi = (pk − Ak ) (δik − θεij ∂j Ak ) ,
mṗi = (pk − Ak ) ∂i Ak .
(17)
In the symmetric gauge Ai = − B2 εij xj , these equations yield
1
mẍi = εij ẋj B (1 + θB)
4
(18)
and in the Landau gauge (A1 , A2 ) = (0, Bx1 ) we get
mẍ1 = ẋ2
B
,
1 + θB
mẍ2 = −ẋ1 B (1 + θB) .
(19)
Thus, it seems that for θ 6= 0 the equations of motion (resulting from the minimally coupled
1
~ )2 ) depend on the chosen gauge, and that they differ from those
(~p − A
Hamiltonian H = 2m
obtained in equation (16) (by introducing the magnetic field into the Poisson brackets).
This point can be elucidated by evaluating the Poisson bracket between the components
~
of the kinematical momentum p~ − A:
{p1 − A1 , p2 − A2 }PB = ∂1 A2 − ∂2 A1 + {A1 , A2}PB ≡ F12 .
(20)
The resulting expression for the field strength F12 involves a quadratic term {A1, A2 }PB
which is characteristic for non-Abelian gauge field theories. The field F12 appears in the
equation of motion for the velocity vi ≡ ẋi : indeed, by differentiating equations (18) and
(19), we obtain the equation of motion v̈i + ω 2vi = 0 where the frequency ω is defined by
|F12 |
B (1 + 14 θB) for the symmetric gauge
ω≡
, with F12 =
B
for the Landau gauge .
m
8
Thus, the gauge potentials Ai = − B2 εij xj and (A1 , A2 ) = (0, Bx1 ) do not define the same
noncommutative field strength F12 . We will encounter a completely analogous situation
in NCQM (sections 6.2 and 7) and we will discuss in that context the non-Abelian gauge
transformations, the introduction of a constant θ-independent field strength and the comparison between the different approaches to such a magnetic field.
4
NCQM: representations of the algebra
In the sequel of the text, we generally consider a charge e = 1 and a system of units
such that ~ ≡ 1 ≡ c. We always assume that the canonical operators Xi and Pj operate on
L2 (R2 , dx1 dx2 ) as in the standard Schrödinger representation, i.e. Xi acts as multiplication
by xi and Pj = ~i ∂j .
4.1
Representations of the algebra
If the canonical limit (or so-called commutative limit) (θ, B) → 0 exists and is considered, the quantum algebra (1) reduces to the Heisenberg algebra (2) so that the operators
X̂i , P̂j should reduce to the operators Xi , Pj satisfying (2). Thus, it is natural to look
for representations of the algebra (1) for which the operators X̂i , P̂j are operator-valued
functions of the real parameters θ, B and of the operators Xi , Pj .
To determine such representations, one generally considers a linear transformation
~ P~ ) 7→ (X,
~ˆ P~ˆ ) whose transformation matrix depends on the noncommutativity pa(X,
rameters, e.g. see references [24, 37]. In the following, we discuss different representations
of (1) and in section 4.2 we will come back to the existence of the canonical limit.
(i) Case θ = 0 : A simple particular case of (1) is given by the Landau system discussed
in section 2: then, we have the algebra
[X̂1 , X̂2 ] = 0 ,
[P̂1 , P̂2 ] = iB1l ,
[X̂i , P̂j ] = iδij 1l
(21)
and different gauge choices (A1 (~x ), A2 (~x )) with ∂1 A2 − ∂2 A1 = B provide different representations of the algebra (21), all of which have the form
X̂i = Xi ,
~ ) with ∂1 A2 − ∂2 A1 = B .
P̂j = Πj ≡ Pj − Aj (X
(22)
~ A
~ ′ giving rise to the same magnetic field B are related by a gauge
Vector potentials A,
~ 7→ A
~′ = A
~ + ∇α(
~ X
~ ) which goes along with a unitary transformation of
transformation A
′
operators (and states): X̂i = U X̂i U −1 and P̂j′ = U P̂j U −1 with U = eiα . In fact [38], all
representations of (21) are unitarily equivalent6 to the representation (22).
6
For the reasons indicated in the footnote to section 2, one should exponentiate the unbounded selfadjoint operators X̂i , P̂j so as to obtain Weyl’s form of commutation relations: the classification theorem
for the representations actually refers to this form of the commutation relations.
9
(ii) Case B = 0 : A completely analogous situation is the case where the spatial coordinates do not commute whereas the momenta commute:
[X̂1 , X̂2 ] = iθ1l ,
[P̂1 , P̂2 ] = 0 ,
[X̂i , P̂j ] = iδij 1l
(23)
Then,
X̂j = Xj − Ãj (P~ )
P̂i = Pi ,
with
∂ Ã1 ∂ Ã2
−
=θ
∂P2
∂P1
(24)
provides a representation of the algebra (23). In fact, in the present context it is appropriate
to consider Schrödinger’s momentum space representation for the canonical operators Pi
and Xj : the operators P̂i and X̂j then act on wave functions ψ̃ ∈ L2 (R2 , dp1 dp2 ) according
to
∂ ψ̃
− Ãj ψ̃ .
(25)
P̂i ψ̃ = pi ψ̃ ,
X̂j ψ̃ = i
∂pj
The parameter θ then measures the noncommutativity of the ~p-space covariant derivatives.
Obviously, the vector field (Ã1 , Ã2 ) in the ~p-plane is analogous to a vector potential in the
~x-plane, and θ to a magnetic field. For a given θ, different choices of the vector field
~ˆ (just as different choices of the vector
(Ã1 , Ã2 ) may be viewed as different “gauges” for X
potential (A1 , A2 ) define different gauges for the covariant derivatives P̂j in equation (22)).
∂ α̃
(~p ) induces a unitary transformation of X̂i and P̂j
A change of gauge, Ãj 7→ Ã′j = Ãj + ∂p
j
~
by means of the operator V = eiα̃(P ) . Analogously to case (i), all representations of (23)
are unitarily equivalent to the representation (25).
(iii) General case : If we combine the noncommutativities (21) and (23), we obtain the
algebra
[X̂1 , X̂2 ] = iθ1l ,
[P̂1 , P̂2 ] = iB1l ,
[X̂i , P̂j ] = iδij 1l ,
(26)
which can be represented by combining particular realizations of (21) and (23), e.g. [20]
X̂1 = X1
X̂2 = X2 + θP1
P̂1 = P1 + BX2
P̂2 = P2 .
(27)
While the ‘minimal’ representation (27) amounts to considering the “Landau gauge” for
~ˆ we can also choose the “symmetric gauge” for X
~ˆ if θ 6= 0 [24]:
X,
θ
P2
2a
θ
P1
X̂2 = aX2 +
2a
X̂1 = aX1 −
where
a ∈ R∗ ,
c=
P̂1 = cP1 + dX2
(28)
P̂2 = cP2 − dX1 ,
√
1
(1 ± κ ) ,
2a
10
d=
√
a
(1 ∓ κ ) ,
θ
(29)
~ˆ P~ˆ ) in
with κ as defined in (11): κ = 1 − Bθ. The expressions (27) and (28) specifying (X,
~ P~ ) are invertible if and only if the functional determinant
terms of (X,
#
"
~ˆ P~ˆ )
∂(X,
=κ
(30)
det
~ P~ )
∂(X,
does not vanish, i.e. if and only if B 6= 1/θ. As we noticed in section 3.1.2, this result
reflects the linear Darboux theorem which holds in classical mechanics if the symplectic
matrix defining the Poisson brackets of coordinates and momenta is invertible.
For B = 1/θ, the canonical limit (θ, B) → 0 does not exist for the algebra (26) nor for
the given representations: we will come back to this case in section 4.2. At this point, we
only note that the representation (27) is reducible for B = 1/θ since the operators (27) then
leave invariant the linear space of functions of the form ψ(x1 , x2 ) = f (x1 ) exp [i(λ − xθ1 )x2 ].
If the coordinates and momenta are gathered into a phase space vector
t
t
ˆ
~u ≡ (ûI )I=1,...,4 = X̂1 , P̂1 , X̂2 , P̂2 ,
then the commutation relations (26) take the form [ûI , ûJ ] = iM̂IJ 1l where M̂ is a constant
antisymmetric 4×4 matrix. As discussed in references [21, 39], the matrix M̂ can be blockdiagonalized by means of an O(4) matrix R (i.e. Rt R = 1l). For B 6= 1/θ, the components
of the transformed phase space vector ~u = Rt~uˆ can be rescaled so as to obtain new phase
~ˆ P~ˆ ) satisfying
space variables X1 , P1 , X2 , P2 which satisfy CCR’s. Thus, for variables (X,
the algebra (26) with B 6= 1/θ, there exists an invertible linear transformation to canonical
~ P~ ). Since all representations of the CCR’s are unitarily equivalent, this also
variables (X,
holds for the representations of the algebra (26) if B 6= 1/θ.
To conclude, we note that the algebra (26) can be decoupled by virtue of appropriate
linear combinations of generators [20, 22]: for θ 6= 0, we may consider the change of
~ˆ P~ˆ ) 7→ (X,
~ˆ K
~ˆ ) given by
generators (X,
1
εjk X̂k
for j = 1, 2 ,
θ
which yields a direct sum of two algebras of the same form:
κ
[X̂1 , X̂2 ] = iθ1l ,
[K̂1 , K̂2 ] = −i 1l ,
[X̂i , K̂j ] = 0 .
θ
K̂j = P̂j −
(31)
(32)
If B = 1/θ (i.e. κ = 0), one concludes from (32) that K̂1 and K̂2 commute with all
generators of the algebra (i.e. they belong to the center of the algebra and represent
Casimir operators). By Schur’s lemma, they are c-number operators for any irreducible
representation, i.e. K̂i = λi 1l with λi ∈ R. In this case, the only nontrivial commutator is
the one involving X̂1 and X̂2 which may be rewritten as
1
X̂2 ,
θ
i.e. we have canonically conjugate variables in one dimension.
[X̂1 , Π̂] = i1l
with Π̂ =
11
(33)
4.2
About the relation between the parameters θ and B
Let us start from noncommuting coordinates in the plane and assume as before that the
~ˆ and P~ˆ is canonical [17]:
commutator between X
[X̂1 , X̂2 ] = iθ1l ,
[X̂i , P̂j ] = iδij 1l ,
[P̂1 , P̂2 ] ∝ i1l .
(34)
Furthermore, let us assume that the considered representation of this algebra is irreducible.
For the linear combination K̂j = P̂j − θ1 εjk X̂k introduced in equation (31), we have
[K̂i , X̂j ] = 0 ,
[K̂1 , P̂1 ] = 0 = [K̂2 , P̂2 ]
1
[K̂1 , P̂2 ] = [P̂1 , P̂2 ] − i 1l = −[K̂2 , P̂1 ]
θ
and [K̂1 , K̂2 ] = [P̂1 , P̂2 ] − i
[P̂1 , P̂2 ] = i
(35)
1
1l. Thus, there are two particular cases:
θ
1
1l
θ
or
[P̂1 , P̂2 ] = i B 1l with B 6=
1
.
θ
For the first case, it follows from equations (35) that the operators K̂1 and K̂2 commute
~ˆ and P~ˆ . By Schur’s lemma they are constant multiples of the identity operator so
with X
ˆ
~ˆ e.g. for vanishing constants: P̂j = 1 εjk X̂k . (For instance, for
that P~ is a function of X,
θ
the irreducible representation (28) we have K̂1 = 0 = K̂2 if B ≡ 1/θ.) Henceforth, the only
independent commutator is then given by [X̂1 , X̂2 ] = iθ1l and describes a 1-dimensional
canonical system: we have a degeneracy of the representation of the quantum algebra in
the sense that the representation of the algebra of the X̂i alone becomes irreducible [22, 25].
In view of this result, the phase space with B = 1/θ might be referred to as a “degenerate”
or “singular” noncommutative space. For this space, the canonical limit θ → 0 does not
~ˆ and P~ˆ in terms of canonical variables X
~ and
exist and one cannot express the variables X
P~ by virtue of an invertible transformation – see equation (30).
By contrast, in the second case the canonical limit (θ, B) → 0 can be taken and there
~ˆ P~ˆ ) and (X,
~ P~ ), e.g. the
exists an invertible transformation relating the variables (X,
transformation (27). Particular instances of this case are given by θ = 0, B 6= 0 and
θ 6= 0, B = 0.
5
5.1
NCQM systems: generalities
About Hamiltonians
~ P~ ) of standard quantum mechanics in Rd can be generalized to
A given Hamiltonian H(X,
~ˆ P~ˆ ) of NCQM by replacing the canonical coordinates and momenta by
a Hamiltonian H(X,
non-canonical ones. (The ordering problem for the variables X̂1 , . . . , X̂d is to be discussed in
12
the next subsection.) The so-obtained Hamiltonian is still assumed to act on standard wave
functions ψ ∈ L2 (Rd , dd x). This action is clearly defined once an explicit representation
~ˆ and P~ˆ in terms of X
~ and P~ is given. The independence of the spectral properties of
of X
~ˆ P~ˆ ) from the representation chosen for X,
~ˆ P~ˆ will be addressed in section 5.3 below.
H(X,
~ˆ P
~ˆ ) can be determined perturbatively by treating the noncomThe spectrum of H(X,
mutativity parameters θij as small perturbative parameters. In some particular cases, the
~ˆ P~ˆ ) can be determined exactly by algebraic methods without referring
spectrum of H(X,
~ˆ P~ˆ ) in terms of (X,
~ P~ ).
to any explicit representation of (X,
5.2
Remarks on the ordering problem
In two-dimensional NCQM with position operators satisfying [X̂1 , X̂2 ] = iθ1l, one can
consider different ordering prescriptions to define an operator V (X̂1 , X̂2 ) associated to
a classical function V (x1 , x2 ), the most popular one being the Weyl ordering, i.e. the
symmetrization prescription. After introducing the operator â = √12θ (X̂1 + iX̂2 ) and its
adjoint which operators satisfy [â, ↠] = 1l, one can also define the normal ordering (all
operators â to the right) or the anti-normal ordering (all operators ↠to the right) [12, 25].
For instance, for a central potential given by a power law, one has at the classical level
N
N
V (~x 2 ) ≡ ~x 2N = x21 + x22 ∼ (2θ)N ↠â
( N = 1, 2, . . . ) ,
where the tilde symbol indicates that some ordering prescription needs to be specified for
the last expression at the quantum level. E.g. for the so-called holomorphic polarization à
la Bargmann-Fock [12, 25], one chooses the anti-normal order:
N
↠â anti-normal order ≡ âN (↠)N = (↠a + 1l) (↠â + 21l) · · · (↠â + N1l) .
~ˆ 2 ), which is radially symAlternatively, the operator-valued function V (r̂ 2 ) (with r̂ 2 ≡ X
metric, may simply be assumed to act on Hilbert space as multiplication by the classical
function V (~x 2 ).
For observables which may be probed experimentally, one can (and one should) resort
to the experimental results for choosing the “good” quantization scheme [40, 41]. However
in the case where no precise experimental data are available (as it appears to be the case for
the models of NCQM), it is in general impossible to find a preferred quantization scheme
(although one may invoke some extra criteria favoring certain choices [42]).
5.3
Properties of energy spectra
Before looking at some specific examples, it is worthwhile to determine some general prop~ˆ P~ˆ ).
erties of the spectrum of a given Hamiltonian H(X,
~ˆ P~ˆ ) in terms of canonical variables
As emphasized earlier, the reparametrization of (X,
~ P~ ) does not represent a unitary transformation since it modifies the commutators.
(X,
13
~ˆ P~ˆ ) is not unitarily equivalent to its commutative counHenceforth the Hamiltonian H(X,
~ P~ ). By way of consequence, their spectra are in general different.
terpart H(X,
As argued in section 4.1, all representations of the quantum algebra (26) are unitarily
~ˆ P~ˆ ) and (X
~ˆ ′ , P~ˆ ′ ), we have
equivalent for B 6= 1/θ, i.e. for any two representations (X,
X̂i′ = U X̂i U −1 , P̂j′ = U P̂j U −1 with U unitary. Thus, the corresponding Hamiltonians are
also unitarily equivalent,
~ˆ ′ , P~ˆ ′ ) = UH(X,
~ˆ P~ˆ ) U −1 ,
H(X
(36)
which ensures that the energy spectrum does not depend on the representation that has
been chosen for the quantum algebra.
By way of example, we briefly elaborate on the particular case where B = 0, i.e. on
the quantum algebra (23). This algebra is realized by the operators (24) which act on
L2 (R2 , dp1 dp2 ) as in equation (25). Different representations of the quantum algebra, i.e.
~ˆ are related by a gauge transformation described by a function α̃(~p ):
different gauges for X
∂ α̃
Ãj 7→ Ã′j = Ãj + ∂p
. The induced transformation of X̂i and P̂j is a unitary transformation:
j
X̂i′ = U X̂i U −1 and P̂j′ = U P̂j U −1 with U = eiα̃(~p ) . The wave function ψ̃(p1 , p2 ) transforms
~ˆ P~ˆ ), i.e.
according to ψ̃ 7→ ψ̃ ′ = U ψ̃ = eiα̃(~p ) ψ̃ and the eigenvalue equation for H(X,
~ˆ P~ˆ ) does
H ψ̃ = E ψ̃, becomes H ′ ψ̃ ′ = E ψ̃ ′ . Thus, the spectrum of the Hamiltonian H(X,
~ˆ i.e. on the representation which is considered for
not depend on the gauge chosen for X,
the quantum algebra.
6
NCQM systems: operatorial approach
In this section, we discuss the standard operatorial approach to NCQM. The star product
approach (“deformation quantization”) will be treated in section 7 while the path integral
approach is to be commented upon elsewhere [26].
The noncommutative Landau problem (i.e. a particle in the noncommutative plane coupled to a constant magnetic field which is perpendicular to this plane [20]) can be treated
along the lines of its commutative counterpart. Thus, two equivalent approaches can be
considered: either the Hamiltonian is expressed in terms of the canonical momentum and
a vector potential describing the magnetic field, or the Hamiltonian has the form of a free
particle Hamiltonian, but involving a momentum whose components do not mutually commute, their commutator being given by the magnetic field strength. The latter approach
will be described in the first subsection. We note that this formulation (which does not
involve a gauge potential) cannot be generalized to non-constant magnetic fields since the
Jacobi identities for the commutator algebra are violated for θ 6= 0 and B depending on
~ˆ (see section 3.1.3).
X
In subsection 6.2, we will discuss the approach relying on a vector potential and we will
see that the noncommutativity of the configuration space variables then leads to expressions
14
which are characteristic for non-Abelian Yang-Mills (YM) theories. A comparison between
the different approaches to magnetic fields in NCQM will be made in section 8.
~ and P~ are assumed
As before and unless otherwise stated, the canonical operators X
2
2
to act on L (R , dx1 dx2 ) in the standard manner.
6.1
Commutation relations involving a constant magnetic field
ˆ
For a particle of unit mass and charge, we consider the Hamiltonian H = 21 P~ 2 which
depends on momentum variables P̂j which do not commute with each other, but rather
satisfy the algebra (26):
[X̂1 , X̂2 ] = iθ1l ,
[P̂1 , P̂2 ] = iB1l ,
[X̂i , P̂j ] = iδij 1l .
The spectrum of H may be determined by mimicking the reasoning presented in equations
(7)-(9): the calculation and the final result
1
En = |B| (n + )
with n ∈ {0, 1, 2, . . . }
(37)
2
do not involve the parameter θ which means that the energy levels are identical in standard
quantum mechanics (θ = 0) and in NCQM (θ 6= 0) 7 . This result reflects the fact that the
Hamiltonian does not depend on the coordinates X̂1 , X̂2 so that its spectrum is insensitive
to the value of the commutator [X̂1 , X̂2 ]. However, there are observable quantities which
do not coincide in standard quantum mechanics and in NCQM, e.g. [20] the density of
1 B . The latter diverges at the critical point B = 1/θ.
states per unit area ρ =
2π 1 − Bθ A detailed study of the physical observables in the noncommutative Landau system is
presented in reference [43].
6.2
Minimal coupling to a (possibly non-constant) magnetic field
We start [44] from the algebra describing the noncommutativity of configuration space,
[X̂1 , X̂2 ] = iθ1l ,
[P̂1 , P̂2 ] = 0 ,
[X̂i , P̂j ] = iδij 1l
(38)
~ˆ )2 . Here, A
~ˆ is a vector field depending on the
− eA
~ˆ ≡ A(
~ X
~ˆ ), with some ordering prescription for the
non-canonical operators X̂1 , X̂2 , i.e. A
variables X̂1 and X̂2 . The operator X̂i is to be thought of as a function of the canonical
ˆ
~
operators Xj and Pk (see equation (24) for a general expression), while P~ = P~ = −i∇.
ˆ
~ =
From relations (38), it follows that the components of the kinematical momentum Π
ˆ
~ˆ satisfy
P~ − eA
and from the Hamiltonian H =
ˆ
1 ~
(P
2m
[Π̂1 , Π̂2 ] = ieF̂12 1l ,
with F̂12 ≡ ∂1 Â2 − ∂2 Â1 − ie [Â1 , Â2 ]
7
(39)
Concerning this result, it is worthwhile recalling from section 3.2.1 that the classical equations of
motion for θ = 0 and θ 6= 0 are also identical.
15
where ∂j ≡ ∂ /∂xj . We note that F̂12 has the same expression as the field strength in nonAbelian YM theories. Since X̂i depends on P~ (according to (24)), one has [X̂i , Π̂j ] = iδij 1l
plus additional terms (specified in equation (60) below within the star product approach).
Since the vector potential depends on the noncommuting variables X̂i , it is natural to
~ˆ Thus, the wave function ψ(~x )
assume that the gauge transformations also depend on X.
~ˆ
transforms as ψ 7→ ψ ′ = Û ψ with Û = eiλ(X ) . By requiring the covariant derivative Π̂i ψ
to transform in the same manner as ψ, one finds
i
Â′i = Û Âi Û −1 + Û ∂i Û −1
e
′
−1
F̂ij = Û F̂ij Û .
(40)
Henceforth, the consideration of noncommuting coordinates in configuration space yields
transformation laws for the gauge fields that are characteristic for a non-Abelian gauge
~ˆ it is
theory. If the noncommutative field strength F̂12 is constant (i.e. independent of X),
gauge invariant by virtue of equation (40).
~ˆ
More specifically, let us now consider the symmetric gauge for A,
B̄
B̄
ˆ
~ = − X̂2 , X̂1 ,
A
(41)
2
2
where B̄ is a constant (which might depend on θ). Substitution of (41) into (39) yields
e
(42)
F̂12 = Λ̄B̄ ,
with Λ̄ = 1 + θB̄ .
4
If B̄ depends on θ and on a θ-independent constant B according to
2 √
B̄ = B̄(B; θ) ≡
1 + eθB − 1
(43)
eθ
e
= B (1 + θB) + O(θ2 ) ,
4
then relation (42) implies that the noncommutative field strength F̂12 is a θ-independent
constant: F̂12 = B. In this case, we have the same algebraic setting as in subsection 6.1:
~ˆ 2 and depends on the non-canonical momentum comthe Hamiltonian reads as H = 1 Π
2m
ponents Π̂j which satisfy [Π̂1 , Π̂2 ] = ieB1l (where B is a θ-independent constant). Thus,
the spectrum of H is the same as in ordinary quantum mechanics: En = |eB|
(n + 21 ) with
m
n = 0, 1, . . . .
~ˆ = 0, B X̂1
The same result can be obtained by considering the Landau gauge A
which yields the same constant field strength: F̂12 = B. By contrast to the star product
formalism described in section 7.2 below, the approach described in this subsection does
~ˆ (i.e. a specific representation of (X,
~ˆ P~ˆ ) in terms
not fix afore-hand a specific gauge for X
~ P~ )) and it works in a simple way for any choice of gauge potential A
~ˆ describing a
of (X,
~ˆ and Π
~ˆ have the
constant magnetic field strength F̂12 . The commutation relations for X
form (60) and they allow for a treatment of variable magnetic fields.
16
6.3
“Exotic” approach
We note that there exists another description of the NC Landau system, namely the socalled “exotic” model [25, 15, 23]. This approach is based on the Poisson algebra (12),
i.e. it amounts to modifying the commutators (26) by a factor 1/κ. Applications of the
present model to the fractional quantum Hall effect and to the vortex dynamics in a thin
superfluid 4 He film are discussed in reference [25].
7
NCQM systems: star product approaches
Consider the CCR’s (2) of standard quantum mechanics in two dimensions:
[X1 , X2 ] = 0 = [P1 , P2 ] ,
[Xi , Pj ] = iδij 1
for i, j ∈ {1, 2} .
In the operatorial approach, one works with functions of the noncommuting operators
Xi , Pj and with the ordinary product of such functions. Alternatively, one can consider
functions f (~x, ~p ) depending on the ordinary commuting coordinates xi , pj and multiply
these functions by a noncommutative product, namely the so-called star product. This
formalism, initiated by Weyl and Wigner [45], developped by Groenewold and Moyal [46]
and further generalized in the sequel [47, 48, 49] allows for a phase space description of
quantum mechanics. This autonomous approach to quantum mechanics is also referred to
as deformation quantization - see [49, 50, 51] for a nice summary and overview. Here, we
only note that the original formalism of Groenewold and Moyal assumes Weyl ordering of
the noncommuting variables.
The procedure of deformation quantization can be generalized in different manners to
NCQM as described (in its simplest form) by the algebra
[X̂1 , X̂2 ] = iθ1l ,
[P̂1 , P̂2 ] = 0 ,
[X̂i , P̂j ] = iδij 1l
for i, j ∈ {1, 2} .
(44)
The obvious generalization consists of defining a star product for phase space functions
f (~x, ~p ) which implements all of the commutation relations (44). In this formulation, a
quantum state is also a function on phase space, namely the so-called Wigner function.
Such a phase space formulation of NCQM has been considered by some authors, e.g. [21, 52].
Another approach [16], to be treated in detail in this section, proceeds along the lines
of the star product formulation of field theory on noncommutative space [53]: here one
only introduces a star product in configuration space in order to implement the
noncommutativity of the x-coordinates, i.e. the first of relations (44). By definition, the
Groenewold-Moyal star product of two smooth functions V and ψ depending on ~x is
a series in θ given by
2
i
1
1 i ij
θ ∂i V ∂j ψ +
θij θkl ∂i ∂k V ∂j ∂l ψ + . . . ,
(45)
V ⋆ψ =Vψ+
1! 2
2! 2
which implies that
[x1 ⋆, x2 ] ≡ x1 ⋆ x2 − x2 ⋆ x1 = iθ .
17
For this formulation of NCQM, quantum states are described as usual by wave functions
ψ(~x ) on configuration space and the momentum variables are also assumed to act as usual
on these functions, i.e. Pj = −i ∂/∂xj (as in the representation (24) of the commutation
relations (44)).
Before considering a particle subject to a magnetic field, we investigate the simpler case
of a particle in a scalar potential.
7.1
Scalar potentials
In the standard formulation of ordinary quantum mechanics, the potential energy V acts
on the wave function ψ ∈ L2 (R2 , dx1 dx2 ) as an operator of multiplication: ψ 7→ V · ψ.
Accordingly, in the star product formulation of NCQM, the potential energy V acts on the
wave function ψ by the star product (45). With the help of the Fourier transform, one can
check [16, 14] that the expression (45) can be rewritten as
1
with X̂i ≡ Xi − θij Pj .
2
~ˆ ) ψ(~x )
(V ⋆ ψ) (~x ) = V (X
(46)
~ˆ ) is to be considered as an operator acting on the wave function
Here, the quantity V (X
ψ ∈ L2 (R2 , dx1 dx2 ). In particular, we have
(xi ⋆ ψ) (~x ) = X̂i ψ(~x ) .
(47)
Thus, we recover the representation
X̂i = Xi −
1 ij
θ Pj ,
2
P̂j = Pj ,
~ˆ
of the algebra (44), i.e. the “symmetric gauge” for X.
Obviously, equation (46) corresponds to a particular ordering prescription: the righthand-side is defined in terms of the left-hand-side which is unambiguous. One can verify
that this prescription is indeed the one of Weyl: by spelling out (xi xj ) ⋆ ψ with the help of
(45), one finds
1
(48)
((xi xj ) ⋆ ψ) (~x ) = (X̂i X̂j + X̂j X̂i ) ψ(~x ) .
2
In summary: For NCQM based on the algebra (44), there are two equivalent approaches
(at least for sufficiently regular potential functions): first, the operatorial approach in
which the Hamiltonian is expressed in terms of the operators P̂j ≡ Pj ≡ −i∂j and in terms
of the non-canonical operators X̂i and, second, the star product approach in which the
Hamiltonian is written in terms of Pj ≡ −i∂j and real coordinates xi acting on the wave
function by the Groenewold-Moyal star product (47). The latter formulation relies on the
Weyl ordering of the variables X̂i and corresponds to the choice of the symmetric gauge for
~ˆ see equations (46)-(48). (For a different ordering prescription, namely normal ordering
X,
(hence a different star product), we refer to [54].)
18
1
p~ 2 + V (~x ) acting on wave functions by the star product,
For the Hamiltonian H = 2m
the probabilistic interpretation proceeds along the usual lines, all products being replaced
by star products. Indeed, the time derivative of the probability density ρ ≡ ψ̄ ⋆ ψ can be
determined by using the Schrödinger equation
H ⋆ ψ = i~ ∂t ψ ,
with H =
1 2
~p + V (~x )
2m
and its complex conjugate (taking into account the fact that V ⋆ ψ = ψ̄ ⋆ V for the realvalued potential V ). Thus, one obtains the continuity equation
ρ = ψ̄ ⋆ ψ
∂t ρ + div ~ = 0 , with
~
~ − (∇
~ ψ̄) ⋆ ψ ] ,
~ = 2mi
[ ψ̄ ⋆ ∇ψ
R
which implies that the total probability R2 d2 x ψ̄ ⋆ ψ is a conserved quantity.
7.2
Magnetic fields
Let us now consider a particle moving in the noncommutative plane subject to a (possibly
non-constant) magnetic field which is perpendicular to this plane. The present section
describes two approaches to this problem which are both based on the star product in
configuration space and on the introduction of a vector potential (i.e. a U(1) gauge field).
The first approach solely relies on the star product [16, 55] while the second one supplements
the star product with the so-called Seiberg-Witten map [56, 57, 58, 59]. The results
obtained for the particular case of a constant magnetic field (i.e. the Landau system) will
be compared with those obtained for this system in section 6.
7.2.1
Gauge potentials
We use the following notation. The fields of NCQM (which depend on the coordinates
~x = (x1 , x2 ) and which are multiplied by each other by means of the star product) are
denoted by a ‘check’8 e.g. ψ̌, Ǎi , λ̌, . . . .
A local U(1) gauge transformation of the wave function ψ̌ (describing a particle of
charge e) is given by
ψ̌ 7−→ ψ̌ ′ = Ǔλ̌ ⋆ ψ̌ ,
with Ǔλ̌ = e⋆ieλ̌ ,
(49)
where all products of λ̌ in the exponential Ǔλ̌ are star products. The inverse of Ǔλ̌ is
determined by the relation Ǔλ̌ ⋆ Ǔλ̌−1 = 1. The covariant derivative of ψ̌, as defined by
Ďi ψ̌ = ∂i ψ̌ −ieǍi ⋆ ψ̌, transforms in the same way as ψ̌ if the vector potential Ǎi transforms
according to
i
(50)
Ǎi 7−→ Ǎ′i = Ǔλ̌ ⋆ Ǎi ⋆ Ǔλ̌−1 + Ǔλ̌ ⋆ (∂i Ǔλ̌−1 ) .
e
8
We do not use the standard notation [60], i.e. a ‘hat’, in order to avoid any confusion with the hatted
operators X̂i and P̂j considered in the earlier sections.
19
The commutator of the covariant derivatives determines the magnetic field strength:
[Ďi ⋆, Ďj ]ψ̌ = −ieF̌ij ⋆ ψ̌ ,
with F̌ij = ∂i Ǎj − ∂j Ǎi − ie [Ǎi ⋆, Ǎj ] .
(51)
The latter transforms according to
F̌ij 7−→ F̌ij′ = Ǔλ̌ ⋆ F̌ij ⋆ Ǔλ̌−1 .
(52)
Thus, for the Abelian gauge group U(1), the noncommutativity of the star product induces
expressions that are characteristic for non-Abelian Yang-Mills theories [16, 8, 61]. For the
two-dimensional configuration space that we consider here for simplicity, the only nonvanishing component of the tensor F̌ij is F̌12 ≡ B∗ . By contrast to the field strength
Fij = ∂i Aj − ∂j Ai of an ordinary U(1) gauge theory, the field strength F̌ij is not gauge
invariant, but transforms covariantly as in equation (52). Instead of the fundamental
representation for the matter field (that we consider here and for which the covariant
derivative reads as Ďi ψ̌ = ∂i ψ̌ − ieǍi ⋆ ψ̌), one can also consider the anti-fundamental
or the adjoint representation (for which we have, respectively, Ďi ψ̌ ≡ ∂i ψ̌ + ieψ̌ ⋆ Ǎi and
Ďi ψ̌ ≡ ∂i ψ̌ − ie[Ǎi ⋆, ψ̌]).
The Schrödinger equation Ȟ ⋆ ψ̌ = i∂t ψ̌ with
X
1 ~
~ˇ )2 ≡ 1
(Pi − eǍi ) ⋆ (Pi − eǍi )
(53)
(P − eA
Ȟ =
⋆
2m
2m i
is invariant under the gauge transformations (49),(50) with a gauge parameter λ̌ which
does not depend on time. Let us now consider the case where the field strength F̌ij is
constant. This case is quite particular since it follows from (52) that such a field F̌ij is
gauge invariant.
Let us choose [16, 55] the symmetric gauge
B̄
B̄
ˇ
~
(54)
A(~x ) = − x2 , x1 ,
2
2
where the real constant B̄ parametrizes the magnetic field. From (51), we infer that the
associated field strength is given by
e
F̌12 = Λ̄B̄ ,
with Λ̄ = 1 + θB̄ .
(55)
4
Henceforth, we presently have expressions of the same form as in the operatorial approach
– compare (54) with (41) and (55) with (42).
The star products can be disentangled by virtue of relation (47):
e
(56)
(Pi − eǍi ) ⋆ ψ̌ = Λ̄ Pi − Ǎi (~x ) ψ̌ .
Λ̄
By applying this relation once more, one finds [16] that Ȟ ⋆ ψ̌ (with Ǎi given by (54)) reads
as
Ȟ ⋆ ψ̌ = H∗ ψ̌ ,
(57)
20
with
2
1 ~
~ˇ x ) , where m∗ = Λ̄−2 m , e∗ = Λ̄−1 e .
P − e∗ A(~
2m∗
Here, we put a lower index ∗ on the quantities which have been modified due to the
consideration of star products (i.e. the quantities H∗ , m∗ and e∗ ).
The Hamiltonian H∗ has the same form as the corresponding Hamiltonian in ordinary
quantum mechanics (the only difference being a redefined mass and charge which depend
on θ). As a consequence of this fact and of the relation ∂1 Ǎ2 − ∂2 Ǎ1 = B̄, the spectrum of
H∗ is given, in analogy to expression (9), by
H∗ =
1
|e∗ B̄|
(n + )
m∗ c
2
|eF̌12 |
1
=~
(n + )
mc
2
En = ~
with n ∈ {0, 1, . . . } .
(58)
These energy levels are the same as in ordinary quantum mechanics, but with the magnetic
field B̄ = ∂1 Ǎ2 − ∂2 Ǎ1 replaced by the noncommutative field strength F̌12 . The latter field
~ˇ depends on
depends on θ according to (55) except for the case where B̄ (and thereby A)
θ in such a way that F̌12 is a θ-independent constant B: if relation (43) holds, i.e. if
2 √
1 + eθB − 1
B̄ = B̄(B; θ) ≡
eθ
e
= B (1 + θB) + O(θ2 ) ,
4
(59)
then equation (51) yields F̌12 = B. The symmetric gauge (54) with B̄ given by (59) is
~ˇ = (0, Bx1 ).
gauge equivalent to the Landau gauge A
In summary: There are two different view-points for the noncommutative Landau system described in terms of a vector potential and the star product. The first view-point is
that the variable B̄ in the vector potential (54) is a θ-independent constant: this leads to
a θ-dependent energy spectrum (En ∝ |Λ̄B̄|) which differs from the one that we obtained
by the approach of section 6.1 (En ∝ |B̄|) where we considered a free Hamiltonian and a
commutator of momenta determined by the constant magnetic field: [P̂1 , P̂2 ] ≡ iB̄1l. The
second view-point consists of regarding the noncommutative field strength F̌12 as the central variable, thereby considering this field as a θ-independent constant: F̌12 = B. In this
case, the vector potential (54) depends on θ according to (59). This implies that the energy
spectra obtained, respectively, by the approach of this section and the one of section 6.1
(with [P̂1 , P̂2 ] ≡ iB1l) coincide with each other: En ∝ |B|. This spectrum also coincides
with the one of ordinary quantum mechanics.
Remarks:
(i) As was already pointed out in reference [16], the argumentation (56),(57) relies on
~ˇ (e.g. equation (56) does not hold in a Landau-type gauge).
the symmetric gauge (54) for A
This result is related to the fact that the star product amounts to choosing the symmetric
21
~ˆ – see equation (47). Henceforth, the spectrum for the minimally coupled
gauge for X
Hamiltonian can only be determined by simple algebraic arguments if the symmetric gauge
~ˇ Other gauge choices require more complicated
is chosen as well for the vector potential A.
computations.
(ii) The gauge invariance of the ⋆-eigenvalue equation Ȟ ⋆ ψ̌ = En ψ̌ is ensured by construction. The ordinary eigenvalue equation Ȟ∗ ψ̌ = En ψ̌ is invariant under ordinary gauge
~ˇ and ψ̌, the relationship between the different gauge transformations
transformations of A
being the subject of the Seiberg-Witten map discussed in the next subsection.
(iii) If a scalar potential V̌ is added to the Hamiltonian (53), the function V̌ ⋆ ψ̌
transforms in the same manner as ψ̌ under a local U(1) gauge transformation Ǔλ̌ , if the
potential V̌ transforms according to V̌ 7→ V̌ ′ = Ǔλ̌ ⋆ V̌ ⋆ Ǔλ̌−1 [59]. The example of a
~ = −∇V
~ is treated in reference [55] and applied to the Hall effect
constant electric field E
and the Aharonov-Bohm effect.
(iv) If we introduce the components of the kinematical momentum, Π̌k ≡ −i Ďk =
−i∂k − eǍk , we obtain the ⋆-commutation relations
[x1 ⋆, x2 ] = iθ
[Π̌1 ⋆, Π̌2 ] = ie F̌12 ,
with F̌12 = ∂1 Ǎ2 − ∂2 Ǎ1 − ie [Ǎ1 ⋆, Ǎ2 ]
[xi ⋆, Π̌j ] = i δij − e [xi ⋆, Ǎj ] .
(60)
By contrast to the commutation relations (26) involving a constant magnetic field B, i.e.
[X̂1 , X̂2 ] = iθ1l ,
[P̂1 , P̂2 ] = iB1l ,
[X̂i , P̂j ] = iδij 1l ,
(61)
we may presently have a non-constant magnetic field, but the algebra (60) contains some
extra terms on the right-hand side which are due to the fact that x1 and x2 do not ⋆commute. These additional terms ensure the validity of the Jacobi identities for the algebra
~ˇ In particular, the Jacobi identities hold for a vector
(60) for any vector potential A.
potential describing a variable magnetic field by contrast to the algebra (61) for which
~ˆ . Yet, if the field strength F̌12 is not
these identities do not hold if B depends on X
constant, it is not gauge invariant which renders its physical interpretation as a magnetic
field somewhat unclear. As we will see in the next subsection, this issue can be settled by
considering the Seiberg-Witten map which relates the noncommutative gauge fields and
parameters to their commutative counterparts in a specific way.
7.2.2
Gauge potentials and Seiberg-Witten map
In the previous subsection, we saw that the consideration of star products in the Schrödinger
equation coupled to a U(1) gauge field leads to the emergence of a non-Abelian gauge
structure in NCQM which is not present in ordinary quantum mechanics. The situation
is completely analogous in relativistic field theory when passing from an ordinary U(1)
gauge theory to a noncommutative U(1) gauge theory. In the latter context, Seiberg and
22
Witten [60] have introduced a map between both theories which relates noncommutative
gauge fields and parameters to their commutative counterparts in such a way that gauge
equivalent field configurations are mapped into each other. By virtue of this map, both
theories can be considered as equivalent and the action functional or Hamiltonian of the
noncommutative U(1) theory can be expressed in terms of ordinary U(1) gauge fields and
the noncommutativity parameters θij (which may be viewed as a constant “background
field”).
In this section, we will discuss this so-called Seiberg-Witten map within the context
of quantum mechanics [56, 57, 58, 59]. We use the same notation as in the previous
subsection, i.e. the fields of NCQM (which depend on the coordinates ~x = (x1 , x2 ) and
which are multiplied with each other by means of the star product) are denoted by a
‘check’. We also spell out the coupling constant e which is usually absorbed into the fields.
By definition, the Seiberg-Witten map is a mapping Ai 7→ Ǎi (A) and λ 7→ λ̌(λ, A) (the
‘checked’ fields being power series in the noncommutativity parameters θij = θεij ) which
satisfies
(62)
Ǎi (A) + δ̌λ̌ Ǎi (A) = Ǎi (A + δλ A) .
Here, δ̌λ̌ Ǎi = Ďi λ̌ ≡ ∂i λ̌ − ie [Ǎi ⋆, λ̌ ] and δλ Ai = ∂i λ describe infinitesimal U(1) gauge
transformations in NCQM and in ordinary quantum mechanics, respectively. Equation
(62) is solved [60] by
e kl
θ Ak (∂l Ai + Fli ) + O(θ2 )
2
e
λ −
7 → λ̌(λ, A; θ) = λ − θij Ai ∂j λ + O(θ2 ) ,
2
Ai 7−→ Ǎi (A; θ) = Ai −
(63)
and we have analogous maps [62] for the matter field ψ and the field strength of A:
e ij
θ Ai ∂j ψ + O(θ2 )
2
7 → F̌ij (A; θ) = Fij + eθkl Fik Fjl + O(θ2 ) .
−
ψ 7−→ ψ̌(ψ, A; θ) = ψ −
Fij
(64)
Here, the expression for F̌ij follows from the defining relation (51). The fact that λ̌ not
only depends on λ, but also on A implies there is no well-defined mapping between the
gauge group {Uλ = eieλ } of ordinary gauge theory and the gauge group {Ǔλ̌ = e⋆ieλ̌ } of
noncommutative gauge theory [60].
For a constant magnetic field B ≡ F12 , one has the following exact (i.e. valid to all
orders in θ) result for the noncommutative field strength F̌12 ≡ B̌ which is determined by
the Seiberg-Witten map [60]:
B
.
(65)
B̌ =
1 − eθB
Just as the ordinary magnetic field B can be described by a symmetric gauge field
configuration, Ai = − 21 Bεij xj (with i ∈ {1, 2}), the Seiberg-Witten field strength B̌ ≡ F̌12
23
can be described by a symmetric field configuration [59]:
1
Ǎi (A; θ) = − B̄(B; θ) εij xj ,
2
1
√
−1
1 − eθB
3
= B(1 + eθB) + O(θ2 ) .
4
2
with B̄(B; θ) =
eθ
(66)
Indeed, substitution of this expression into F̌ij , as defined by relation (51), yields the result
(65). The relationship with the expressions in the previous subsection can easily be worked
out9 .
To discuss the Landau problem, we again proceed as in the previous subsection, see
equations (53)-(58). Thus, we consider the Hamiltonian
Ȟ =
X
1 ~
~ˇ )2 ≡ 1
(Pi − eǍi ) ⋆ (Pi − eǍi ) ,
(P − eA
⋆
2m̌
2m̌ i
(67)
acting on the Seiberg-Witten field ψ̌. In expression (67), the Seiberg-Witten gauge field Ǎi
describing a constant magnetic field is assumed to be given by the symmetric gauge (66).
The mass parameter has been denoted by m̌. Very much like the Seiberg-Witten fields, it
may be viewed as a function of θ satisfying m̌(θ = 0) = m.
By using relation (47) to disentangle the star products, we get
Ȟ ⋆ ψ̌ = HSW ψ̌ ,
with
HSW
2
1 ~
ˇ
~
=
P − eSW A(~x ) ,
2mSW
where mSW = Λ̄−2 m̌ , eSW = Λ̄−1 e .
(68)
The Hamiltonian HSW (which depends on the constant “background field” θ) has the same
form as the corresponding Hamiltonian in ordinary quantum mechanics. As a consequence
of this fact and of the relation ∂1 Ǎ2 − ∂2 Ǎ1 = B̄, the spectrum of HSW is given by
En = ~
|eSW B̄|
1
|eB̌|
1
(n + ) = ~
(n + )
mSW c
2
m̌c
2
with n ∈ {0, 1, . . . } .
If we assume that
m
,
1 − eθB
then it follows from (65) that the energy levels read as
m̌ =
En = ~
|eB|
1
(n + )
mc
2
with n ∈ {0, 1, . . . } .
9
(69)
(70)
In fact, in the present section we wrote B ≡ F12 rather than B ≡ F12 since the variable B was
used in section 7.2.1 with a different meaning. As in subsection 7.2.1, we presently have F̌12 = Λ̄B̄ with
Λ̄ = 1 + 4e θB̄. If B = B (1 + eθB)−1 , we recover the expressions of subsection 7.2.1: B̄ as a function of B
is then given by equation (59) and F̌12 = B.
24
This result coincides with the one of ordinary quantum mechanics (for a particle of mass m
and charge e coupled to the constant magnetic field B = ∂1 A2 − ∂2 A1 ). In this sense, both
theories are physically equivalent. Note that the singularity B = (eθ)−1 of the parameters
B̄ and m̌ does not manifest itself in the spectrum (70).
A somewhat different treatment of the Landau problem within the Seiberg-Witten
framework (leading to the same result for the spectrum) is presented in reference [59].
In the latter treatment, relation (69) follows from the requirement that the Hamiltonians
HSW (θ) and HSW (0) are related by a unitary transformation which ensures that the physics
remains invariant under a change in θ. The approach discussed in this subsection also allows
to incorporate scalar potentials (see reference [57] for applications to the Hall effect).
We note that substitution of the symmetric gauge Ai = − 21 Bεij xj into ψ̌, as given by
equation (64), leads to
ψ̌(ψ, A; θ) = ψ +
eθB
~ ψ + O(θ2 ) .
~x · ∇
4
Here, the first order term in θ amounts to a θ-dependent scale transformation of ψ, which
shows that the Seiberg-Witten map (64) is not unitary [59]. This map may eventually
be unitarized, see references [57, 59]. Different aspects of the Seiberg-Witten map (like
singularities or ambiguities in the parametrization) are discussed in the work [63].
In summary: The approach of Seiberg-Witten to NCQM amounts to considering star
products, and to assuming that the fields of NCQM (i.e. ψ̌, Ǎi , λ̌, . . . ) and the coupling
constants of NCQM (i.e. eSW and mSW ) are specific functions of those occurring in ordinary
~ˆ and
quantum mechanics. The star product is tantamount to the symmetric gauge for X
thereby this approach to a constant magnetic field only works in a simple way if the
~ˇ
symmetric gauge is also chosen for A.
8
Summary and concluding remarks
Magnetic fields in NCQM – Summary: For [X̂1 , X̂2 ] = iθ1l, the coupling to a constant
ˆ
magnetic field can either be described by considering a free Hamiltonian H = 1 P~ 2 and a
2m
non-trivial commutator [P̂1 , P̂2 ] = ieB1l, or by considering a trivial commutator [P̂1 , P̂2 ] = 0
and a Hamiltonian involving a vector potential. For the latter case, we presented three
approaches (subsections 6.2, 7.2.1 and 7.2.2) all of which lead to a noncommutative field
strength. The vector potential or field strength appearing in these approaches can be
expressed in various ways in terms of the constant θ-independent field B. By choosing this
dependence on B in an appropriate way, one recovers the spectral results of the former
formalism (where [P̂1 , P̂2 ] = ieB1l). The issue of gauge invariance of the energy spectrum
is unproblematic in the operatorial formulation (section 6.2) and somewhat subtle in the
other approaches.
The three approaches based on vector potentials lead to a non-canonical commutator
between the coordinates and the kinematical momentum – see equation (60). Thereby,
25
they allow for a treatment of a non-constant magnetic field to be identified with the
noncommutative field strength F12 . However, the physical interpretation is somewhat
subtle due to the fact that the field F12 is not gauge invariant in this case.
Concluding remarks: In these notes, we discussed some general properties of NCQM
while limiting ourselves to two dimensions and to the case of a canonical commutator
between X̂i and P̂j : [X̂i , P̂j ] = iδij 1l. The case of higher dimensions and of a more general
algebra are to be addressed elsewhere [26] along with other physical systems involving
both scalar and vector potentials. In that context, we will also review other methods of
quantization like the path integral approach to NCQM.
Acknowledgments
It is a great pleasure to thank H. Grosse, P. Exner, M. Kibler and S. Richard for valuable
discussions. F.G. thanks M. Rausch de Traubenberg, M. Sampaio, J. Dittrich and V. Rivasseau for the kind invitations to deliver talks on the present subject at the conferences
“Rencontres Alsaciennes Math-Physique” (Strasbourg), “International Meeting on Topics in Quantum Field Theory” (Belo Horizonte), “Operator Theory in Quantum Physics”
(Prague) and “Non-commutative Geometry and Physics” (Orsay). We also acknowledge
the discussions on noncommutative field theories with A. Deandrea, C. Gaw̧edzki, J. Grimstrup, N. Mamoudi and P. Schuck.
26
A
Noncommuting coordinates in the truncated Landau problem (Peierls’ substitution)
In the following, we will show that noncommuting coordinates naturally appear in the
Landau problem in the limit of a very strong magnetic field [6, 12]. Since the separation of
Landau levels (9) is of order B/m, a large magnetic field B is equivalent to a small mass
m and the limit m → 0 amounts to a restriction to the lowest Landau level (LLL). Before
presenting this limiting procedure in a rigorous manner, we provide a simple heuristic
argument [12, 2]. The classical dynamics of the Landau problem (supplemented by a weak
scalar potential V (x1 , x2 ) describing impurities in the plane) is described by the Lagrangian
1 ˙2 e ~ ˙
m~x + A · ~x − V (~x )
2
c
1
eB
= m(ẋ21 + ẋ22 ) +
x1 ẋ2 − V (x1 , x2 ) ,
2
c
Lm =
(71)
where we have chosen the Landau gauge (A1 , A2 ) = (0, Bx1 ). In the limit m → 0, we are
left with
eB
L0 =
x1 ẋ2 − V (x1 , x2 ) ,
c
i.e. an expression of the form pq̇ − H0 (p, q) with a pair of canonically conjugate variables
(p, q) = ( eB
x , x2 ) and a Hamiltonian H0 = V . Upon quantization, the classical variables
c 1
X̂1 and X̂2 satisfying CCR’s, [ eB
X̂1 , X̂2 ] = −i~ 1l, i.e.
become operators eB
c
c
[X̂1 , X̂2 ] = −i
~c
1l .
eB
(72)
Thus, the restriction of the particle dynamics to the LLL amounts to introducing noncommuting coordinates in the plane which satisfy the algebra (72), the reduced Hamiltonian
H0 = V (X̂1 , X̂2 ) describing this dynamics being given by the potential depending on the
noncommuting coordinates. This so-called Peierls’ substitution means that the effect of
an impurity (as described by the potential V in the Lagrangian (71)) on the eigenstate of
the LLL can be evaluated to lowest order by computing the eigenvalues ǫn of the effective
Hamiltonian H0 = V (X̂1 , X̂2 ), where the commutator of X̂1 and X̂2 is proportional to 1/B.
More explicitly, if V (X̂1 , X̂2 )|ni = ǫn |ni, then the energy levels for the quantum system
|eB|
) in the approximation
associated to (71) are given by En = 12 ~ωB + ǫn (with ωB ≡
mc
of strong B and weak V [6, 12, 2]. (Incidentally, the result (72) coincides – up to a sign
– with the commutator [X̂1 , X̂2 ] that appears in the algebra (26) at the singular point
κ ≡ 1 − ~ce Bθ = 0.)
To make the previous argumentPmore precise [64] (see also [2, 54, 65]), we consider
2
e
1
2
the Hamiltonian operator H = 2m
j=1 (Pj − c Aj ) as written in the symmetric gauge.
Recall that the eigenstate |n, ki of H associated to the n-th Landau Rlevel En = ~ωB (n + 12 )
is infinitely degenerate: k ∈ R. The projection operator Pn = R dk |n, ki hn, k| onto
27
the corresponding
eigenspace can be expressed as follows in terms of the Hamiltonian
P
E
P
[64]:
H= ∞
n=0 n n
sin (n + 21 ) π (Hn − 1l)
1
4
,
with Hn ≡
H.
Pn =
π(2n + 1)
(Hn − 1l) (Hn + 1l)
En
Rather than truncating the theory to the LLL, we may consider more generally the theory
obtained by cutting off at an energy E with
PNEN ≤ E < EN +1 , where N ∈ {0, 1, 2, . . . }
is fixed [54, 65, 64]. The operator ΠN = n=0 Pn then projects onto the corresponding
subspace of Hilbert space H. In particular, for any operator B on H, the expression
B̂ ≡ ΠN BΠN
represents the truncation of B. The authors of reference [64] determined the commutators
of the truncated canonical operators:
~c
(N + 1) PN
eB
eB
(N + 1) PN
[P̂1 , P̂2 ] = −i~
4c
1
[X̂i , P̂j ] = i~ δij [1 − (N + 1)] PN .
2
[X̂1 , X̂2 ] = −i
(73)
In particular, the first equation implies that the LLL matrix elements of [X̂1 , X̂2 ] are given
by
~c
h0, k| [X̂1, X̂2 ] |0, k ′ i = −i
h0, k|0, k ′i
eB
in agreement with the result (72) suggested by an heuristic argument. Thus, the set of
equations (73) gives a precise mathematical meaning to the procedure of ‘projection to
LLL’. For N → ∞, one recovers the canonical commutators [65].
The modifications to (73) brought about an interaction potential Vλ (x21 + x22 ) are discussed in reference [66]. In general, the commutator [X̂1 , X̂2 ] then depends on the parameter λ characterizing the potential Vλ .
28
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